Question

what system of equations represents the graph?
a
$y = -\frac{1}{2}x + 10$
$y = -2x$

b
$y = -2x + 10$
$y = -\frac{1}{2}x$

c
$y = -2x + 10$
$y = -\frac{1}{3}x$

Explanation:

Step1: Analyze the first line (steeper line)

The steeper line has a y - intercept of 10 (since it crosses the y - axis at (0,10)) and a slope. Let's calculate the slope. We can use two points, say (0,10) and (5,0) (since it crosses the x - axis at (5,0)). The slope \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{0 - 10}{5 - 0}=\frac{- 10}{5}=-2\). So the equation of this line is \(y=-2x + 10\) (using the slope - intercept form \(y=mx + b\), where \(m=-2\) and \(b = 10\)).

Step2: Analyze the second line (less steep line)

The less steep line passes through the origin \((0,0)\), so its y - intercept \(b = 0\). Let's find its slope. We can use two points, say (0,0) and (4,-2). The slope \(m=\frac{-2-0}{4 - 0}=\frac{-2}{4}=-\frac{1}{2}\). So the equation of this line is \(y =-\frac{1}{2}x\) (using \(y=mx + b\) with \(m =-\frac{1}{2}\) and \(b = 0\)).

Step3: Match with the options

Looking at the options, option B has the equations \(y=-2x + 10\) (for the steeper line) and \(y =-\frac{1}{2}x\) (for the less steep line), which matches our calculations.

Answer:

B. \(y = - 2x+10\); \(y=-\frac{1}{2}x\)